1 2 ( 3 4 ) 2 + 7 8 \frac{1}{2}\left(\frac{3}{4}\right)^2+\frac{7}{8} 2 1 ( 4 3 ) 2 + 8 7 Write Your Answer In Simplest Form.

Mar 8, 2025 by ADMIN 132 views

$Simplifying the Expression: $\frac{1}{2}\left(\frac{3}{4}\right)^2+\frac{7}{8}$$

Understanding the Problem

The given expression involves the addition of two fractions, one of which is a squared fraction. To simplify this expression, we need to follow the order of operations (PEMDAS) and perform the necessary calculations.

Breaking Down the Expression

The expression can be broken down into two parts:

$\frac{1}{2}\left(\frac{3}{4}\right)^2$
$\frac{7}{8}$

Simplifying the First Part

To simplify the first part, we need to follow the order of operations and perform the exponentiation first.

$\left(\frac{3}{4}\right)^2 = \frac{3^2}{4^2} = \frac{9}{16}$

Now, we can multiply this result by $\frac{1}{2}$ .

$\frac{1}{2}\left(\frac{9}{16}\right) = \frac{1 \times 9}{2 \times 16} = \frac{9}{32}$

Simplifying the Second Part

The second part of the expression is already simplified, so we can move on to the next step.

Adding the Two Parts

Now that we have simplified both parts of the expression, we can add them together.

$\frac{9}{32} + \frac{7}{8}$

To add these fractions, we need to find a common denominator. The least common multiple (LCM) of 32 and 8 is 32.

$\frac{9}{32} + \frac{7}{8} = \frac{9}{32} + \frac{7 \times 4}{8 \times 4} = \frac{9}{32} + \frac{28}{32}$

Now, we can add the numerators.

$\frac{9 + 28}{32} = \frac{37}{32}$

Final Answer

The simplified expression is $\frac{37}{32}$ .

Conclusion

In this article, we simplified the given expression by following the order of operations and performing the necessary calculations. We broke down the expression into two parts, simplified each part, and then added them together to get the final answer.

Tips and Tricks

When simplifying expressions, always follow the order of operations (PEMDAS).
When adding fractions, find a common denominator and add the numerators.
When multiplying fractions, multiply the numerators and denominators separately.

Final Thoughts

Simplifying expressions is an essential skill in mathematics, and it requires attention to detail and a thorough understanding of the order of operations. By following the steps outlined in this article, you can simplify even the most complex expressions and arrive at the correct answer.

Frequently Asked Questions

In this article, we will address some of the most common questions related to simplifying expressions. Whether you're a student, a teacher, or simply someone who wants to improve their math skills, this Q&A section is for you.

Q: What is the order of operations (PEMDAS)?

A: The order of operations (PEMDAS) is a set of rules that tells us which operations to perform first when simplifying expressions. PEMDAS stands for:

Parentheses: Evaluate expressions inside parentheses first.
Exponents: Evaluate any exponential expressions next (e.g., 2^3).
Multiplication and Division: Evaluate any multiplication and division operations from left to right.
Addition and Subtraction: Finally, evaluate any addition and subtraction operations from left to right.

Q: How do I simplify a fraction?

A: To simplify a fraction, you need to find the greatest common divisor (GCD) of the numerator and denominator. The GCD is the largest number that divides both numbers evenly. Once you've found the GCD, you can divide both the numerator and denominator by it to simplify the fraction.

Q: What is the difference between a numerator and a denominator?

A: In a fraction, the numerator is the number on top, and the denominator is the number on the bottom. For example, in the fraction 3/4, 3 is the numerator and 4 is the denominator.

Q: How do I add fractions with different denominators?

A: To add fractions with different denominators, you need to find a common denominator. The common denominator is the least common multiple (LCM) of the two denominators. Once you've found the common denominator, you can convert both fractions to have that denominator and then add them.

Q: Can I simplify an expression with multiple fractions?

A: Yes, you can simplify an expression with multiple fractions by following the same steps as before. First, simplify each fraction individually, and then combine them using the order of operations.

Q: What is the difference between a simplified expression and a simplified fraction?

A: A simplified fraction is a fraction that has been reduced to its simplest form by dividing both the numerator and denominator by their greatest common divisor. A simplified expression, on the other hand, is an expression that has been simplified by applying the order of operations and combining like terms.

Q: How do I know when an expression is simplified?

A: An expression is simplified when it has been reduced to its simplest form by applying the order of operations and combining like terms. You can check if an expression is simplified by looking for any remaining parentheses, exponents, or fractions that can be simplified further.

Q: Can I use a calculator to simplify expressions?

A: Yes, you can use a calculator to simplify expressions, but it's always a good idea to double-check your work by hand to ensure that the calculator is giving you the correct answer.

Q: What are some common mistakes to avoid when simplifying expressions?

A: Some common mistakes to avoid when simplifying expressions include:

Forgetting to apply the order of operations
Not simplifying fractions before combining them
Not combining like terms
Not checking for any remaining parentheses, exponents, or fractions that can be simplified further

Q: How can I practice simplifying expressions?

A: You can practice simplifying expressions by working through math problems, either on your own or with a teacher or tutor. You can also use online resources, such as math websites or apps, to practice simplifying expressions.

Q: What are some real-world applications of simplifying expressions?

A: Simplifying expressions has many real-world applications, including:

Calculating costs and prices
Determining the area and perimeter of shapes
Finding the volume of objects
Solving problems in physics, engineering, and other fields

Q: Can I use simplifying expressions to solve word problems?

A: Yes, you can use simplifying expressions to solve word problems. By applying the order of operations and combining like terms, you can simplify complex expressions and arrive at a solution to the problem.

Q: How can I use simplifying expressions to improve my math skills?

A: You can use simplifying expressions to improve your math skills by practicing regularly and applying the concepts to real-world problems. By mastering the art of simplifying expressions, you can become a more confident and proficient math student.